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Keywords:
weighted Hardy space; operator; Gaussian estimate; duality; product space
Summary:
Let $L$ be a non-negative self-adjoint operator acting on $L^2({\mathbb R}^n)$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_r$ weight on ${\mathbb R}^n\times {\mathbb R}^n$, $1<r<\infty $. In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H^{p}_{L,w}({\mathbb R}^n\times {\mathbb R}^n)$, $0<p\leq 1$ associated to $L$. Based on the atomic decomposition, we show the dual relationship between $H^{1}_{L,w}({\mathbb R}^n\times {\mathbb R}^n)$ and ${\rm BMO}_{L,w}({\mathbb R}^n\times {\mathbb R}^n)$.
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